Reprojection and backprojection methods and algorithms for implementation thereof

ABSTRACT

Methods for projecting and backprojecting rays with respect to pixels/detector bins to attenuate/eliminate high-frequency artifacts, are disclosed. The first two methods are adaptations of pixel-driven and ray-driven linear interpolation techniques respectively. In these techniques, the window or shadow of each pixel/bin is dynamically adjusted and projected onto the detector bin/pixel to eliminate gaps between the shadows. This allows the effect of each pixel on a given detector bin (or vice versa) to be appropriately weighted. A third is a distance-driven technique wherein the transitions of the pixels and the detector bins are respectively projected onto a common axis. This allows a determination of the contribution of each of the pixels/bins for each of the bins/pixels with lower computation time and improved artifact free images.

BACKGROUND OF THE INVENTION

[0001] The present invention relates generally to the processes ofreprojection-backprojection, and more specifically, toreprojection-backprojection techniques/algorithms that includes newinterpolation and data access schemes that result in better speed, lowerartifacts, lower noise and higher spatial resolution than existingtechniques.

[0002] In computed tomography, the operation that transforms anN-Dimension image into an N-Dimension set of line integrals is calledthe forward projection or reprojection. The most evident example of thisoperation is the physical process that generates an X-ray image of anobject. After logarithmic conversion, an X-ray image is wellapproximated as the line integral projection of the distribution of theobject's linear attenuation coefficient. In practice, a forwardprojector is required for tomographic simulations or when performingiterative reconstruction.

[0003] The transpose operation is called backprojection. This is used infiltered backprojection and in iterative reconstruction, which form thebulk of today's reconstruction algorithms.

[0004] Many methods for reprojection and backprojection exist. In onemethod each X-ray beam is represented by a line and the intersectionlength of each line with each pixel is used as weight factor. Anothertechnique performs linear interpolation between two pixels for each rowor column that the X-ray beam intersects (see FIG. 1). The latter twomethods are ray-driven methods.

[0005] In the projection case, all projection lines are looped over, andfor each projection line the image weighting and summing image pixelvalues are run through in order to approximate a ray-integral. Thebackprojection is defined as the transpose operation: the weight factorsremain the same, but the detector values are weighted and assigned tothe image pixels.

[0006] Another technique is the pixel-driven approach, which istypically used in filtered backprojection (see FIG. 2). All image pixelsare looped over, and for each image pixel a line is drawn connecting thesource and the image pixel. The intersection of that line with thedetector array is then determined. Linear interpolation is performedbetween the two detector values nearest to the intersection point andthe result is assigned to the image pixel. The reprojection operation isdefined as the transpose operation. The weights for the left and rightdetector bin are given by $\begin{matrix}\begin{matrix}{\omega_{l} = \frac{d_{r} - d}{d_{r} - d_{l}}} \\{\omega_{r} = \frac{d - d_{l}}{d_{r} - d_{l}}}\end{matrix} & {{Eqn}\quad (1)}\end{matrix}$

[0007] where d is the location of the intersection, d_(r) and d_(l) arethe first detector bin centers to the right and to the left of theintersection.

[0008] Other approaches exist, such as methods based on spherical basicfunctions and methods using nearest-neighbor or no interpolation.

[0009] The reprojection and backprojection operations are acomputationally intensive but essential part of simulation andreconstruction techniques such as those used in CT or the like. Mostexisting approaches can be subdivided into ray-driven and pixel drivenmethods. One drawback to both the ray-driven and pixel driven methodsresides in the fact that they introduce artifacts, the first one (viz.,the ray driven method) in the backprojection and the latter (viz., thepixel driven method) in the reprojection. Another drawback to bothmethods resides in the percentage of the data used in each viewreprojection/backprojection.

[0010] For example, in the case of a ray-driven projection of an imagewith pixels that are much smaller than the detector bin size, only afraction of the pixels contributes to the projection at that angle. Thesame is true for the opposite case of the pixel driven backprojection.In iterative reconstruction, where both a reprojection andbackprojection method are required, a combination of a ray-drivenreprojection and pixel-driven backprojection could be considered tocircumvent previous problems. However, even while this is possible, itis often preferred to use a matched reprojector-backprojector pair. Infact, an important criterion in choosing a reprojector-backprojectorapproach is speed.

[0011] The two main limiting factors on speed are arithmetic complexityand data access time. For the ray-driven approach, the arithmetics isrelatively simple. It is therefore much faster than the pixel drivenapproach for small data sizes. At larger data sizes however, the dataaccess time becomes more important and at this stage the pixel-drivenapproach starts to benefit from its sequential image access time whilethe ray-driven approach more or less accesses the data randomly. For the3D cone-beam case, data sets become even larger and therefore dataaccess time gains importance.

[0012] For further disclosure pertaining to these techniques and thetypes of apparatus which are used in connection therewith, reference maybe had to U.S. Pat. No. 5,848,114 issued on Dec. 8, 1998 in the name ofKawai et al.; U.S. Pat. No. 6,351,514 issued in the name of Besson onFeb. 26, 2002; U.S. Pat. No. 6,339,632 issued in the name of Besson onJan. 15, 2002. The contents of these patents is hereby incorporated byreference thereto.

SUMMARY OF THE INVENTION

[0013] More specifically, a first aspect of the present inventionresides in a method of image processing comprising: projecting pixels ina pixel grid onto a detector having a plurality of bins, or vice versa;dynamically adjusting a dimension of a square window for one of a pixeland a detector bin so that adjacent windows form a continuous shadowover one of the detector bins of the detector and the image pixels; anddetermining the effect of each pixel on each bin of the detector or viceversa.

[0014] A second aspect of the invention resides in a method of imageprocessing comprising: projecting edges of each pixel of a pixel grid,that is intersected by a ray projected from a source to a detector, in apredetermined linear sequence of pixels in the pixel grid, onto apredetermined line that passes through the grid; projecting the edges ofeach bin of a detector onto the predetermined line; and determining thecontribution of each pixel to a bin of the detector array or vice versain accordance with the projections of the pixel edges and the detectorbin edges on the predetermined line.

[0015] A third aspect of the present invention resides in a method ofimage processing comprising: establishing a pixel grid containing imagepixels which are arranged in image rows and columns; continuouslymapping respective transitions between image pixels and detector-bins ofa detector which has detected radiation from a radiation sourcecomprising: projecting detector bin transitions onto a predeterminedline; projecting the pixel transitions onto the predetermined line; andweighting one of the detector bins and pixels with segment lengths onthe predetermined line, based on distances between adjacent projectionsto calculate their contribution.

[0016] A fourth aspect of the present invention resides in a computerreadable medium encoded with a program executable by a computer forprocessing an image, said program being configured to instruct thecomputer to: project pixels in a pixel grid onto a detector having aplurality of bins, or vice versa; dynamically adjust a dimension of asquare window for one of a pixel and a detector bin so that adjacentwindows form a continuous shadow over one of the detector bins of thedetector and the image pixels; and determine the effect of each pixel oneach bin of the detector or vice versa.

[0017] A fifth aspect of the invention resides in a computer readablemedium encoded with a program executable by a computer for processing animage, said program being configured to instruct the computer to:project edges of each pixel of a pixel grid, that is intersected by aray projected from a source to a detector, in a predetermined linearsequence of pixels in the pixel grid, onto a predetermined line thatpasses through the grid; project the edges of each bin of a detectoronto the predetermined line; and determine the contribution of eachpixel to a bin of the detector array or vice versa in accordance withthe projections of the pixel edges and the detector bin edges on thepredetermined line.

[0018] A sixth aspect of the present invention resides in a computerreadable medium encoded with a program executable by a computer forprocessing an image, said program being configured to instruct thecomputer to: establish a pixel grid containing image pixels which arearranged in image rows and columns; continuously map respectivetransitions between image pixels and detector-bins of a detector whichhas detected radiation from a source by: projecting detector bintransitions onto a predetermined line; projecting the pixel transitionsonto the predetermined line; and weighting one of the detector bins andpixels with segment lengths on the predetermined line, based ondistances between adjacent projections to calculate their contribution.

BRIEF DESCRIPTION OF THE DRAWINGS

[0019]FIG. 1 is a schematic representation of a ray-drivenreprojection-backprojection with linear interpolation wherein, for everyrow or column intersected by the projection line, linear interpolationis performed between the two adjacent pixels.

[0020]FIG. 2 is a schematic representation of a pixel-drivenreprojection-backprojection with linear interpolation wherein a lineconnecting source and image pixel determines an intersection with thedetector array and wherein linear interpolation is performed between thetwo adjacent detector bins.

[0021]FIG. 3 is a depiction of a ray-driven backprojection of a uniformview showing the result wherein high-frequency artifacts are introducedbecause some pixels are updated more frequently than their neighbors.

[0022]FIG. 4 is a graphical representation of a pixel-driven projectionof a uniform disk wherein high-frequency artifacts are introducedbecause some detector bins are updated more frequently than theirneighbors.

[0023]FIG. 5 is a schematic depiction of a representation of thepixel-driven linear inter-polation method wherein, due to the irregularoverlap of the projected square windows, some detector bins will seemore contributions than other, resulting in high-frequency oscillations.

[0024]FIG. 6 depicts a pixel-driven linear interpolation method whereinthe width of the square windows is adjusted so that they are alwaysadjacent.

[0025]FIG. 7 depicts a distance-driven reprojector-backprojector whereinboth the detector bin interfaces and the pixel interfaces are mappedonto the x-axis, and wherein the resulting segment lengths are used asweight factors in the projection and backprojection.

[0026]FIG. 8 depicts a distance-driven projector-backprojector providinga closer view of the interlaced pattern of pixel interfaces pi anddetector interfaces di.

[0027]FIG. 9 graphically depicts a distance-driven projection of auniform disk wherein the high-frequency oscillations are entirelyeliminated.

[0028]FIG. 10 is a distance-driven backprojection of a uniform viewwherein the high-frequency artifacts are entirely eliminated.

[0029]FIG. 11 is a graph showing plots of time per backprojection for aSUN E4500 computer versus data size.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0030] In order to better understand the embodiments of the presentinvention a more detailed explanation of the above prior art techniquesis deemed appropriate. In FIGS. 1, 2, 6 and 7 the grid depicts a pixelimage reconstruction grid which is fixed in a three dimensionalcoordinate system, onto which pixels are mapped in accordance with dataacquired in response to a ray being projected from the source to thedetector both (schematically shown). Each of the squares in these gridsdepicts a pixel.

[0031] As noted above, a drawback encountered with both the ray-drivenand the pixel-driven method is that they introduce high-frequencyartifacts, one in the backprojection and another in the reprojection.FIG. 3 shows an example of a ray-driven backprojection of one uniformview. The interference pattern is due to the fact that some pixels areupdated more frequently than other pixels. The artifact problem is worsewhen the pixel size is small compared to the detector bin size, andvanishes when the pixel size is large compared to the detector bin size.

[0032]FIG. 4 graphically shows one sinogram line of a pixel-drivenprojection of a uniform disk. By way of example, in Computed Tomography,a measured data set (sinogram) is made up of a large number of views(projections). Each view corresponds to a measurement with the entiredetector array, so each view in turn consists of a large number ofdetector bins (projection lines). A typical sinogram consists of 1500views/projections of 1000 detector bins/projection lines.

[0033] As mentioned above, the interference pattern is due to the factthat some detector bins are updated more frequently than theirneighbors. Further, the artifact problem is more pronounced when thedetector bin size is small compared to the pixel size, and it vanisheswhen the detector bin size is large compared to the pixel size. In thisinstance the reprojections and backprojections were performed, simply byway of example, with a flat 2D fan-beam geometry, a magnification of1.76, 256×256 pixels, 256 detector bins, 256 views over 360°, and anarbitrary start angle of 126°.

[0034] Another drawback of both methods resides in the data usage ineach view projection/backprojection. Assume, for the sake ofexplanation, a ray-driven projection of an image with pixels that aremuch larger than the detector bin size (see FIG. 5). Only a fraction ofthe pixels contributes to the projection at that angle. Similarly, in apixel-driven backprojection with pixels that are much smaller than thedetector bin size, only a fraction of the detector values are used ineach view. This results in poor noise performance. In iterativereconstruction this may also lead to poor convergence properties.

[0035] A very important criterion in choosing a projector-backprojectorapproach is computation speed. The two main limiting factors oncomputation speed are arithmetic complexity and data access time. Withthe ray-driven approach, the arithmetics is relatively simple. It istherefore faster than the pixel-driven approach for small data sizes. Atlarger data sizes however, the data access time becomes more critical.Under these conditions the pixel-driven approach begins to exhibitdesirable processing speed characteristics due to its inherentsequential image data accessing which reduces access time while theray-driven approach requires a much higher degree of random accessesbecause it jumps over large blocks of data and thus departs from thesequential manner in which the data is stored. This results inprocessing delays.

[0036] For the 3D cone-beam case, however, data sets become even largerand these effects become even more important.

[0037] a) Adaptation of the Pixel-Driven and Ray-DrivenProjector-Backprojector

[0038]FIGS. 5 and 6 respectively demonstrate the features that show theshortcoming encountered with the prior art pixel driven technique and anembodiment of the invention wherein the pixel-driven technique ismodified or adapted to prevent the high-frequency artifacts.

[0039] More specifically, an intersection with the detector array islocated. At the intersection, a Dirac impulse with area equal to thepixel value is assumed. This is convolved with a rectangular window witha width equal to the detector bin size. The weights are obtained byintegrating the result over both adjacent detector bins. This results inthe following formula for the weights: $\begin{matrix}\begin{matrix}\begin{matrix}{\omega_{l} = \frac{d_{m} - ( {d - {( {d_{r} - d_{l}} )/2}} )}{d_{r} - d_{l}}} \\{\omega_{r} = \frac{( {d + {( {d_{r} - d_{l}} )/2}} ) - d_{m}}{d_{r} - d_{l}}}\end{matrix} \\{{d_{m} = \frac{d_{l} + d_{r}}{2}},}\end{matrix} & {{Eqn}\quad (2)}\end{matrix}$

[0040] where d_(m) is the position of the interface centered betweend_(i) and d_(r). This is identical to equation 1, which shows theequivalence of this representation. It is desired, by projecting oneuniform row of pixels, to achieve an essentially uniform projection overthe projected range corresponding to this row (except for the slightlyvarying path length due to the varying location of intersection).However, due to the irregular overlap of the projected square windows,some detector bins will see more contributions than other, resulting inhigh-frequency oscillations.

[0041] This is solved, in accordance with this adapted ray drivenembodiment of the invention, by adjusting the width of the squarewindows or shadows of the pixels so that they are always adjacent and sothat gaps are eliminated and they effectively become continuous. This isillustrated by the gray shadowed areas in FIG. 6 and can be expressedas: $\begin{matrix}\begin{matrix}\begin{matrix}{\omega_{l} = {\max ( {\frac{{\min ( {d_{m},{d + {W/2}}} )} - ( {d - {W/2}} )}{W},0} )}} \\{\omega_{r} = {1 - \omega_{l}}}\end{matrix} \\{{W = {\Delta \quad {p \cdot M \cdot \cos}\quad {\alpha_{d}/\Delta}\quad d}},}\end{matrix} & {{Eqn}\quad (3)}\end{matrix}$

[0042] where W is the new width of the square window, Δp is the pixelsize, Δd is the detector bin size, M is the magnification, and α_(d) isthe angle of the projection line. Cos α_(d) can be pre-calculated if itis approximated by cos α_(d) _(m) . However, the window width W cannotbe larger than the detector bin sized, d_(r)-d_(i), because then it mayoverlap more than 2 detector bins.

[0043] The algorithm could, of course, be generalized to allowoverlapping multiple detector bins using a while-loop for instance.However, this brings about the situation wherein the artifact reductionadvantage does not balance the increase in algorithmic complexity.

[0044] In the adaptation of the pixel driven technique, the dynamicadjustment is applied to the pixels rather than the bins.

[0045] More specifically, a similar argument is made for the artifactsintroduced in the ray-driven backprojection. This results in thefollowing weights for the corrected algorithm: $\begin{matrix}\begin{matrix}\begin{matrix}{\omega_{l} = {\max ( {\frac{{\min ( {p_{m},{p + {W/2}}} )} - ( {p - {W/2}} )}{W},0} )}} \\{\omega_{r} = {1 - {\omega \quad l}}}\end{matrix} \\{{W = {\Delta \quad {{d/M}/\cos}\quad {\alpha_{p}/\Delta}\quad p}},}\end{matrix} & {{Eqn}\quad (4)}\end{matrix}$

[0046] where p is the location of the intersection, and p_(r) and p_(l)are the first pixel centers to the right and to the left of theintersection. However, in this instance, the window width W cannot belarger than the image pixel size, p_(r)-p_(l), because then it mayoverlap more than 2 image pixels.

[0047] The speed of these adapted methods is assumed comparable to theoriginal algorithms. Both adapted methods completely eliminate theartifacts shown in FIGS. 3 and 4, which result with the original methods. . .

[0048] b) Distance-Driven Projection-Backprojection

[0049] The present invention is, in this embodiment, based on acontinuous mapping of the detector array on an image row or column orvice versa and more particularly is based on mapping along the directionof the projection lines. For fast computation, all detector locationsand image locations are projected onto an arbitrarily selected line,which can be, for example, the x- or y-axis of the image.

[0050] With this, the image data are accessed sequentially, similar tothe pixel driven approach, the arithmetic is simple and similar to theray-driven approach, no artifacts are introduced and all data is useduniformly in each view. The new algorithm is amendable forimplementation in both hardware and software, is simple and providesspeed, full data usage which reduces noise, and does not introduceartifacts.

[0051] More specifically, the embodiment of this technique isillustrated in FIG. 7 and is based on a continuous mapping of thedetector array onto an image row (or column) or vice versa, and moreparticularly on mapping along the direction of the projection lines. Forfast computation, the x-axis (or y-axis) is, as mentioned above, used asreference for the relative location of pixels and detector bins. Inorder to define a continuous mapping of image pixels and detector-bins,rather than working with the centers, it is the transitions betweenpixels and between detector bins which are used. First, all detector bintransitions are projected onto the x-axis (or y-axis or an arbitrarilydetermined axis). Next all image rows (or columns) are looped over andthe pixel transitions are projected onto the axis. A value is read fromthe image, weighted with the appropriate segment length defined betweenprojections, and assigned to the detector bin or pixel as the casedemands.

[0052]FIG. 8 shows a more detailed view of the interlaced pattern ofdetector interfaces d_(i), pixel interfaces p_(i), detector valuesd_(ij), and pixel values p_(ij). In this example the contribution of therow under consideration to the ray sums d_(ij) can be written as$\begin{matrix}\begin{matrix}\begin{matrix}{d_{23} = p_{12}} \\{d_{34} = p_{12}}\end{matrix} \\{d_{45} = {\frac{{( {p_{2} - d_{4}} ) \cdot p_{12}} + {( {d_{5} - p_{2}} ) \cdot p_{23}}}{d_{5} - d_{4}}.}}\end{matrix} & {{Eqn}\quad (5)}\end{matrix}$

[0053] while for the backprojection we have $\begin{matrix}\begin{matrix}{P_{12} = \frac{( {{( {d_{2} - p_{1}} ) \cdot d_{12}} + {( {d_{3} - d_{2}} ) \cdot d_{23}} + {( {d_{4} - d_{3}} ) \cdot d_{34}} + {( {p_{2} - d_{4}} ) \cdot d_{34}}} )}{p_{2} - p_{1}}} \\{P_{23} = {\frac{{( {d_{5} - p_{2}} ) \cdot d_{45}} + {( {d_{6} - d_{5}} ) \cdot d_{56}} + {( {p_{3} - d_{6}} ) \cdot d_{67}}}{p_{3} - p_{2}}.}}\end{matrix} & {{Eqn}\quad (6)}\end{matrix}$

[0054]FIG. 9 shows the distance-driven projection of a uniform disk,equivalent to the result of the pixel-driven projection in FIG. 4. Aswill be appreciated, the high-frequency oscillations are, just like withthe adapted pixel-driven projector and with the line-driven projector,entirely eliminated using this technique.

[0055]FIG. 10 shows the distance-driven equivalent of the result of theray-driven backprojection in FIG. 3. Again, the high-frequency artifactsare entirely eliminated with this approach, just like with thepixel-driven backprojector and with the adapted line-drivenbackprojector.

[0056] For a comparison of the performance backprojection was focussedon inasmuch as computation times for projection and backprojection arevery similar. Both the images and the sinograms were chosen to be n×npixels. FIG. 11 is a graph which shows the time required perbackprojection versus data size in using the three different approachesfor a SUN E4500 (10 UltraSPARC-II, 400 Mhz, 8 Mb cache, 10 GB RAM). Forsmall data sizes the arithmetic process forms the bottleneck as all thedata fits in the cache memory. The pixel-driven approach clearlyperforms worst here, while the distance-driven approach comes close tothe ray-driven approach. The same optimization effort has been appliedto all three algorithms. For larger data sets the memory access timebecomes more important, as now the entire image no longer fits in thecache memory. It is only the ray-driven approach that really suffersfrom this, because the memory access is not sequential. This explainsthe slope of the curve for the ray-driven method. For larger data sets,the pixel-driven and distance-driven approaches have the big advantagethat they can be implemented in hardware. The ray-driven one cannot, ashardware hack-projectors cannot generally afford to have access to allof the memory at once.

[0057] The above-disclosed distance-driven projection-backprojectionmethod is summarized below. However, in order to better appreciate thenature of this technique the unamended pixel driven and ray-driventechniques will be firstly outlined.

[0058] Pixel-Driven Technique:

[0059] Address all image pixels (*), and for each image pixel executethe following steps:

[0060] Determine a line connecting the source and the center of theimage pixel.

[0061] Find the intersection of this line with the detector array.

[0062] Determine the two detector bins whose centers are nearest to theintersection.

[0063] For the backprojection: calculate the value at this intersectionby linear interpolation between the two detector bins, and assign thisvalue to the image pixel

[0064] For the (re-)projection: assign the value of the image pixel tothe two detector bins, using the same weights as in the backprojection

[0065] Ray-Driven Technique:

[0066] Address all projection lines (**) (in all views): a projectionline is defined by connecting the source and the center of a detectorbin.

[0067] For each projection line execute the following steps:

[0068] For the (re-)projection: reset the projection sum.

[0069] Address all image rows (***), and for each image row (***) do thefollowing steps:

[0070] Calculate the intersection of the projection line with (thecenterline of) the image row (***).

[0071] Determine the two image pixels in this row (***) whose centersare nearest to the intersection.

[0072] For the (re-)projection:

[0073] calculate the value at this intersection by linear interpolationbetween the two image pixels, and add this value to the projection sum.

[0074] For the backprojection: add the value of the detector bin to thetwo image pixels, using the same weights as in the (re-)projection.

[0075] For the (re-)projection: assign the projection sum to thedetector bin.

[0076] Distance-Driven technique:

[0077] Address all views, and for each view, execute the followingsteps:

[0078] For each detector bin:

[0079] Determine the edges of the detector bin:

[0080] Determine a line by connecting the detector bin edge and thex-ray source

[0081] Calculate the intersection of this line with the x-axis (***)

[0082] This intersection defines the projected detector bin edge

[0083] Address all image rows, and for each image row execute thefollowing steps:

[0084] Address all image pixels in this rows, and for each image pixelexecute the following steps:

[0085] Determine the left and right (***) edges of the image pixel:

[0086] Determine a line by connecting the pixel edge and the x-raysource.

[0087] Calculate the intersection of this line with the x-axis (***)

[0088] This intersection defines the projected pixel edge.

[0089] Make a sorted list of projected detector bin edges and projectedpixel edges

[0090] Start at the first edge that is most left on the x-axis (***),and determine the current pixel and the current detector bin.

[0091] Until the most right edge is reached, execute the followingsteps:

[0092] Determine which is the next edge (****).

[0093] Update the current pixel or the current detector bin.

[0094] Calculate the weight factor as the position of the current edgeminus the position of the previous edge.

[0095] For the (re-)projection: multiply the value of the current imagepixel by the weight factor and add it to the current detector bin.

[0096] For the backprojection: multiply the value of the currentdetector bin by the weight factor and add it to the current image pixel

[0097] Key:

[0098] (*) denotes/pertains to “pixel-driven”

[0099] (**) denotes/pertains to “ray-driven”

[0100] (***) If the orientation of the projection lines is morehorizontal than vertical, then the following conversions are necessary:

[0101] ‘row’ <--> ‘column’

[0102] ‘x-axis’ <--> ‘y-axis’

[0103] ‘left’ <--> ‘bottom’

[0104] ‘right’ <--> ‘top’

[0105] (****) denotes/pertains to “distance-driven” feature

[0106] It should be noted that this summary of the disclosed techniquesis illustrative and not to be taken as specifically limiting the scopeof the invention and that while the preceding disclosure has focussedonly on a limited number of projecting and backprojecting methods theapplication of these techniques is not limited to CT applications. Itshould also be noted that adapting the conventional ray-driven andpixel-driven linear interpolation eliminates high-frequency artifacts,under given restrictive assumptions. The distance-driven method however,entirely eliminates artifacts without any restrictive assumptions, ineach view, all the data contributes uniformly to the resultingprojection or backprojection, and it has favorable computationalproperties.

[0107] Additionally, although methods for a 2D flat-detector fan-beam CTgeometry have been discussed, it will be understood that the methods andconclusions are not limited thereto and that one of skill in this art orone closely related thereto, will appreciate that the concepts areadaptable to other 2D- and 3D (or more)-geometries, including, merely byway of example, PET- and SPECT-geometries.

What is claimed is:
 1. A method of image processing comprising:projecting pixels in a pixel grid onto a detector having a plurality ofbins, or vice versa; dynamically adjusting a dimension of a squarewindow for one of a pixel and a detector bin so that adjacent windowsform a continues shadow over one of the detector bins of the detectorand the image pixels; and determining the effect of each pixel on eachbin of the detector or vice versa.
 2. A method of image processing asset forth in claim 1, wherein dynamic adjustment of a width W of asquare window of a pixel in a pixel-driven image formation technique isdetermined using the equation: $\begin{matrix}\begin{matrix}{\omega_{l} = {\max ( {\frac{{\min ( {d_{m},{d + {W/2}}} )} - ( {d - {W/2}} )}{W},0} )}} \\{\omega_{r} = {1 - \omega_{l}}}\end{matrix} \\{{W = {\Delta \quad {p \cdot M \cdot \cos}\quad {\alpha_{d}/\Delta}\quad d}},}\end{matrix}$

where W is the new width of the square window, Δp is the pixel size, Δdis the detector bin size, M is the magnification, and α_(d) is the angleof the projection line.
 3. A method of image formation as set forth inclaim 1, wherein the dynamic adjustment of the pixel shadows for aray-driven image formation technique is determined using the equation:$\begin{matrix}\begin{matrix}{\omega_{l} = {\max ( {\frac{{\min ( {p_{m},{p + {W/2}}} )} - ( {p - {W/2}} )}{W},0} )}} \\{\omega_{r} = {1 - {\omega \quad l}}}\end{matrix} \\{{W = {\Delta \quad {{d/M}/\cos}\quad {\alpha_{p}/\Delta}\quad p}},}\end{matrix}$

where p is the location of a ray intersection on the detector, and p_(r)and p_(l) are the first pixel centers to the right and to the left ofthe intersection.
 4. A method of image processing as set forth in claim2, wherein the window width W is not larger than the detector bin size.5. A method of image processing as set forth in claim 3, wherein thewindow width W is not larger than image pixel size.
 6. A method of imageprocessing as set forth in claim 2, wherein Cos α_(d) is pre-calculableif it is approximated by cos α_(d) _(m) .
 7. A method of imageprocessing as set forth in claim 3, wherein Cos α_(p) is pre-calculableif it is approximated by cos α_(p) _(m) .
 8. A method of imageprocessing comprising: projecting edges of each pixel of a pixel grid,which is intersected by a ray projected from a source to a detector, ina predetermined linear sequence of pixels in the pixel grid, onto apredetermined line that passes through the grid; projecting the edges ofeach bin of a detector onto the predetermined line; and determining thecontribution of each pixel to a bin of the detector array or vice versain accordance with the projections of the pixel edges and the detectorbin edges on the predetermined line.
 9. A method of image processing asset forth in claim 8, wherein the pixels are rectangular and the step ofprojecting the edges of the pixels onto the predetermined line comprisesprojecting a selected point on a side of the pixel onto thepredetermined line.
 10. A method of image processing as set forth inclaim 8, wherein the predetermined line is an arbitrarily selected line.11. A method of image processing as set forth in claim 8, wherein thepredetermined line is an x-axis.
 12. A method of image processing as setforth in claim 8, wherein the predetermined line is a y-axis.
 13. Amethod of image processing comprising: establishing a pixel gridcontaining image pixels, which are arranged in image rows and columns;continuously mapping respective transitions between image pixels anddetector-bins of a detector which has detected radiation from a sourcecomprising: projecting detector bin transitions onto a predeterminedline; projecting the pixel transitions onto the predetermined line; andweighting one of the detector bins and pixels with segment lengths onthe predetermined line, based on distances between adjacent projections.14. A computer readable medium encoded with a program executable by acomputer for processing an image, said program being configured toinstruct the computer to: project pixels in a pixel grid onto a detectorhaving a plurality of bins, or vice versa; dynamically adjust adimension of a square window for one of a pixel and a detector bin sothat adjacent windows form a continuos shadow over one of the detectorbins of the detector and the image pixels; and determine the effect ofeach pixel on each bin of the detector or vice versa.
 15. A computerreadable medium as set forth in claim 14, wherein dynamic adjustment ofa width W of a square window of a pixel in a pixel-driven imageformation technique is determined using the equation: $\begin{matrix}\begin{matrix}{\omega_{l} = {\max ( {\frac{{\min ( {d_{m},{d + {W/2}}} )} - ( {d - {W/2}} )}{W},0} )}} \\{\omega_{r} = {1 - \omega_{l}}}\end{matrix} \\{{W = {\Delta \quad {p \cdot M \cdot \cos}\quad {\alpha_{d}/\Delta}\quad d}},}\end{matrix}$

where W is the new width of the square window, Δ_(p) is the pixel size,Δd is the detector bin size, M is the magnification, and α_(d) is theangle of the projection line.
 16. A computer readable medium as setforth in claim 14, wherein the dynamic adjustment of the pixel shadowsfor a ray-driven image formation technique is determined using theequation: $\begin{matrix}\begin{matrix}{\omega_{l} = {\max ( {\frac{{\min ( {p_{m},{p + {W/2}}} )} - ( {p - {W/2}} )}{W},0} )}} \\{\omega_{r} = {1 - {\omega \quad l}}}\end{matrix} \\{{W = {\Delta \quad {{d/M}/\cos}\quad {\alpha_{p}/\Delta}\quad p}},}\end{matrix}$

where p is the location of a ray intersection on the detector, and p_(r)and p_(l) are the first pixel centers to the right and to the left ofthe intersection.
 17. A computer readable medium as set forth in claim15, wherein the window width W is not larger than the detector bin size.18. A computer readable medium as set forth in claim 16, wherein thewindow width W is not larger than image pixel size.
 19. A computerreadable medium as set forth in claim 15, wherein Cos α_(d) ispre-calculable if it is approximated by cos α_(p) _(m) .
 20. A computerreadable medium as set forth in claim 16, wherein Cos α_(p) ispre-calculable if it is approximated by cos α_(p) _(m) .
 21. A computerreadable medium encoded with a program executable by a computer forprocessing an image, said program being configured to instruct thecomputer to: project edges of each pixel of a pixel grid, that isintersected by a ray projected from a source to a detector, in apredetermined linear sequence of pixels in the pixel grid, onto apredetermined line that passes through the grid; project the edges ofeach bin of a detector onto the predetermined line; and determine thecontribution of each pixel to a bin of the detector array or vice versain accordance with the projections of the pixel edges and the detectorbin edges on the predetermined line.
 22. A computer readable medium asset forth in claim 21, wherein the pixels are rectangular and the stepof projecting the edges of the pixels onto the predetermined linecomprises projecting a selected point on a side of the pixel onto thepredetermined line.
 23. A computer readable medium as set forth in claim21, wherein the predetermined line is an arbitrarily selected line. 24.A computer readable medium as set forth in claim 21, wherein thepredetermined line is an x-axis.
 25. A computer readable medium as setforth in claim 21, wherein the predetermined line is a y-axis.
 26. Acomputer readable medium encoded with a program executable by a computerfor processing an image, said program being configured to instruct thecomputer to: establish a pixel grid containing image pixels, which arearranged in image rows and columns; continuously map respectivetransitions between image pixels and detector-bins of a detector, whichhas detected radiation from a source by: projecting detector bintransitions onto a predetermined line; projecting the pixel transitionsonto the predetermined line; and weighting one of the detector bins andpixels with segment lengths on the predetermined line, based ondistances between adjacent projections.